On a Use of the Mann-Whitney Statistic

For m = n an equivalent statistic had been proposed and studied earlier by Wilcoxon [2]. The main aim of these studies was to develop a test of the hypothesis that X and Y have the same probability distribution: F = G. More about this test will be reported in section 2. Independently, Haldane and Smith [3] investigated the following problem. In some hereditary conditions, the probability that a member of a sibship has the condition depends partly on his birth rank. Having records of sibships in the order of birth, stating for each individual whether it has or does not have the condition, how does one test for independence of the condition from birth rank? To answer this question, Haldane and Smith constructed a test statistic which is equivalent with the U statistic (1.2). Without an attempt at completeness, we shall give in section 2 a brief survey of the known properties of the U statistic which are of importance for its use in testing hypotheses. The main purpose of the present paper, however, is to discuss another use of this statistic which, while not new, seems to have attracted less attention. Let